A recursive coloring function without $\Pi_3^0$ solution for Hindman theorem

Abstract

We show that there exists a recursive coloring function $c$ such that any $\Pi^0_3$ set is not a solution to $c$ for Hindman's theorem. We also show that there exists a recursive coloring function $c$ such that any $\Delta^0_3$ set is not a solution to $c$ for Hindman's theorem restricted on sum of at most three numbers.